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A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).

%I A169803 #65 Sep 18 2024 10:24:59
%S A169803 1,1,1,1,2,0,1,3,1,0,1,4,3,0,0,1,5,6,1,0,0,1,6,10,4,0,0,0,1,7,15,10,1,
%T A169803 0,0,0,1,8,21,20,5,0,0,0,0,1,9,28,35,15,1,0,0,0,0,1,10,36,56,35,6,0,0,
%U A169803 0,0,0,1,11,45,84,70,21,1,0,0,0,0,0,1,12,55,120,126,56,7,0,0,0,0,0,0
%N A169803 Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n).
%C A169803 T(n,k) = 0 if k <0 or k > n+1-k.
%C A169803 T(n,k) is the number of binary vectors of length n and weight k containing no pair of adjacent 1's.
%C A169803 Take Pascal's triangle A007318 and push the k-th column downwards by 2k-1 places (k>=1).
%C A169803 Row sums are A000045.
%C A169803 From _Emanuele Munarini_, May 24 2011: (Start)
%C A169803 Diagonal sums are A000930(n+1).
%C A169803 A sparse subset (or scattered subset) of {1,2,...,n} is a subset never containing two consecutive elements. T(n,k) is the number of sparse subsets of {1,2,...,n} having size k. For instance, for n=4 and k=2 we have the 3 sparse 2-subsets of {1,2,3,4}: 13, 14, 24. (End)
%C A169803 As a triangle, row 2*n-1 consists of the coefficients of Morgan-Voyce polynomial B(n,x), A172431, and row 2*n to the coefficients of Morgan-Voyce polynomial b(n,x), A054142.
%C A169803 Aside from signs and index shift, the coefficients of the characteristic polynomial of the Coxeter adjacency matrix for the Coxeter group A_n related to the Chebyshev polynomial of the second kind (cf. Damianou link p. 19). - _Tom Copeland_, Oct 11 2014
%C A169803 Antidiagonals of the Pascal matrix A007318 read bottom to top, omitting the first antidiagonal. These are also the antidiagonals (omitting the first antidiagonal) read from top to bottom of the numerical coefficients of the Maurer-Cartan form matrix of the Leibniz group L^(n)(1,1) presented on p. 9 of the Olver paper, which is generated as exp[c. * M] with (c.)^n = c_n and M the Lie infinitesimal generator A218272. Reverse is embedded in A102426. - _Tom Copeland_, Jul 02 2018
%H A169803 Indranil Ghosh, <a href="/A169803/b169803.txt">Rows 0..125, flattened</a>
%H A169803 Pantelis A. Damianou, <a href="http://arxiv.org/abs/1110.6620">On the characteristic polynomials of Cartan matrices and Chebyshev polynomials</a>, arXiv preprint arXiv:1110.6620 [math.RT], 2014.
%H A169803 Emanuele Munarini and Norma Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00616-7">Scattered Subsets</a>, Discrete Mathematics 267 (2003), 213-228.
%H A169803 Emanuele. Munarini and Norma Zagaglia Salvi, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00378-3">On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns</a>, Discrete Mathematics 259 (2002), 163-177.
%H A169803 Emanuele Munarini, <a href="http://dx.doi.org/10.1137/S0895480103432283">A combinatorial interpretation of the Chebyshev polynomials</a>, SIAM Journal on Discrete Mathematics, Volume 20, Issue 3 (2006), 649-655.
%H A169803 Peter J. Olver, <a href="http://www.math.umn.edu/~olver/di_/contact.pdf">The canonical contact form</a>.
%H A169803 James J. Y. Zhao, <a href="https://arxiv.org/abs/2409.08085">Infinite log-concavity and higher order Turán inequality for Speyer's g-polynomial of uniform matroids</a>, arXiv:2409.08085 [math.CO], 2024. See p. 11.
%e A169803 Triangle begins:
%e A169803   [1]
%e A169803   [1, 1]
%e A169803   [1, 2, 0]
%e A169803   [1, 3, 1, 0]
%e A169803   [1, 4, 3, 0, 0]
%e A169803   [1, 5, 6, 1, 0, 0]
%e A169803   [1, 6, 10, 4, 0, 0, 0]
%e A169803   [1, 7, 15, 10, 1, 0, 0, 0]
%e A169803   [1, 8, 21, 20, 5, 0, 0, 0, 0]
%e A169803   [1, 9, 28, 35, 15, 1, 0, 0, 0, 0]
%e A169803   [1, 10, 36, 56, 35, 6, 0, 0, 0, 0, 0]
%e A169803   [1, 11, 45, 84, 70, 21, 1, 0, 0, 0, 0, 0]
%e A169803   [1, 12, 55, 120, 126, 56, 7, 0, 0, 0, 0, 0, 0]
%e A169803   [1, 13, 66, 165, 210, 126, 28, 1, 0, 0, 0, 0, 0, 0]
%e A169803   [1, 14, 78, 220, 330, 252, 84, 8, 0, 0, 0, 0, 0, 0, 0]
%e A169803   [1, 15, 91, 286, 495, 462, 210, 36, 1, 0, 0, 0, 0, 0, 0, 0]
%e A169803   [1, 16, 105, 364, 715, 792, 462, 120, 9, 0, 0, 0, 0, 0, 0, 0, 0]
%e A169803   [1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 0, 0, 0, 0, 0, 0, 0, 0]
%e A169803   [1, 18, 136, 560, 1365, 2002, 1716, 792, 165, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0]
%e A169803   [1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]
%e A169803   ...
%t A169803 T[n_,k_]:= Binomial[n+1-k,k]; Table[T[n,k],{n,0,12},{k,0,n}]//Flatten (* _Stefano Spezia_, Sep 16 2024 *)
%o A169803 (Maxima) create_list(binomial(n-k+1,k),n,0,20,k,0,n); /* _Emanuele Munarini_, May 24 2011 */
%o A169803 (PARI) T(n,k)=binomial(n+1-k,k) \\ _Charles R Greathouse IV_, Oct 24 2012
%Y A169803 Cf. A000045, A000930, A007318, A011973 (another version), A218272.
%Y A169803 All of A011973, A092865, A098925, A102426, A169803 describe essentially the same triangle in different ways. - _N. J. A. Sloane_, May 29 2011
%Y A169803 A172431 and A054142 describe the odd and even lines of the triangle.
%K A169803 nonn,tabl
%O A169803 0,5
%A A169803 Nadia Heninger and _N. J. A. Sloane_, May 21 2010